Random Walks and Brownian Motion Instructor :

Random Walks and Brownian Motion Instructor : Ron Peled

Constructing a sequence of random walks strongly converging to Brownian motion

It is one of the most basic facts in probability theory that random walks, after proper rescaling, converge to Brownian motion. However, Donsker’s classical theorem [Don51] only states a convergence in law. Various results of almost sure convergence exist (see e.g. [KMT75, KMT76] and the references therein) but involves rather intricate relations between the converging sequence of random walks ...

Brownian Bridge Asymptotics for Random Mappings

The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 Aldous-Pitman result on convergence of uniform random mapping walks to reeecting Brownian bridge, an...

Diffusion constants and martingales for senile random walks

We derive diffusion constants and martingales for senile random walks with the help of a time-change. We provide direct computations of the diffusion constants for the time-changed walks. Alternatively, the values of these constants can be derived from martingales associated with the timechanged walks. Using an inverse time-change, the diffusion constants for senile random walks are then obtain...

Balls–in–boxes Duality for Coalescing Random Walks and Coalescing Brownian Motions

We present a duality relation between two systems of coalescing random walks and an analogous duality relation between two systems of coalescing Brownian motions. Our results extends previous work in the literature and we apply it to the study of a system of coalescing Brownian motions with Poisson immigration.